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Math Expert V
Joined: 02 Sep 2009
Posts: 58364
In the xy-plane, triangular region S is bounded by the lines x=0,y=0,  [#permalink]

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Question Stats: 76% (02:43) correct 24% (02:40) wrong based on 219 sessions

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In the xy-plane, triangular region S is bounded by the lines x=0,y=0, and 3x−2y=30. Which of the following points does NOT lie inside region S?

A. (1,−1)
B. (2,−10)
C. (3,−13)
D. (5,−5)
E. (8,−2)

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Manager  B
Joined: 02 Nov 2013
Posts: 72
Location: India
In the xy-plane, triangular region S is bounded by the lines x=0,y=0,  [#permalink]

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This is how went with the problem,

Drawn the shaded region with using three lines x=0, y=0 and 3x-2y=30.

Went back to the answer choices and putting the x values verified the y values. All the points falling in the region should follow equation 3x-2y≤30.

Only answer choice with point (3,−13) does now follow this equation and the value of the equation comes as 35 which is beyond the shaded region.

Hope I am right ?
Attachments Shaded region.png [ 10.76 KiB | Viewed 3545 times ]

Current Student B
Joined: 07 Nov 2016
Posts: 8
Location: India
GMAT 1: 620 Q47 V29 GPA: 3.9
Re: In the xy-plane, triangular region S is bounded by the lines x=0,y=0,  [#permalink]

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I just guessed the answer using the following method,

Using the equation 3x-2y=30, we can find x-intercept and y-intercept
if x=0, then y=-15
if y=0, then x=10
after plotting it on the plane , find the midpoint of these 2 points
(x1+x2/2, y1+y2/2)=(0+10/2,-15+0/2)=>(5,-7.5)

only one answer stands out (3,-13) -> D
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Re: In the xy-plane, triangular region S is bounded by the lines x=0,y=0,  [#permalink]

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Bunuel wrote:
In the xy-plane, triangular region S is bounded by the lines x=0,y=0, and 3x−2y=30. Which of the following points does NOT lie inside region S?

A. (1,−1)
B. (2,−10)
C. (3,−13)
D. (5,−5)
E. (8,−2)

For the point to lie in the specified region
1) Point should be in 4th quadrant (all options are of 4th uadrant)
2) Since 3x−2y=30 i.e. y = (3/2)x-15 so y must be greater than or equal to (3/2)x-15

Checking options
A. (1,−1) -1 > (3/2)(1)-15 Hence, Point lies in the region hence Incorrect Option
B. (2,−10) -10 > (3/2)(2)-15 Hence, Point lies in the region hence Incorrect Option
C. (3,−13) -13 < (3/2)(3)-15 Hence, Point does NOT lie in the region hence CORRECT Option
D. (5,−5) -5 > (3/2)(5)-15 Hence, Point lies in the region hence Incorrect Option
E. (8,−2) -2 > (3/2)(8)-15 Hence, Point lies in the region hence Incorrect Option

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Director  S
Joined: 12 Nov 2016
Posts: 701
Location: United States
Schools: Yale '18
GMAT 1: 650 Q43 V37 GRE 1: Q157 V158 GPA: 2.66
Re: In the xy-plane, triangular region S is bounded by the lines x=0,y=0,  [#permalink]

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GMATinsight wrote:
Bunuel wrote:
In the xy-plane, triangular region S is bounded by the lines x=0,y=0, and 3x−2y=30. Which of the following points does NOT lie inside region S?

A. (1,−1)
B. (2,−10)
C. (3,−13)
D. (5,−5)
E. (8,−2)

For the point to lie in the specified region
1) Point should be in 4th quadrant (all options are of 4th uadrant)
2) Since 3x−2y=30 i.e. y = (3/2)x-15 so y must be greater than or equal to (3/2)x-15

Checking options
A. (1,−1) -1 > (3/2)(1)-15 Hence, Point lies in the region hence Incorrect Option
B. (2,−10) -10 > (3/2)(2)-15 Hence, Point lies in the region hence Incorrect Option
C. (3,−13) -13 < (3/2)(3)-15 Hence, Point does NOT lie in the region hence CORRECT Option
D. (5,−5) -5 > (3/2)(5)-15 Hence, Point lies in the region hence Incorrect Option
E. (8,−2) -2 > (3/2)(8)-15 Hence, Point lies in the region hence Incorrect Option

How do we know that the for the answer choices that y must be greater than the expression 3/2x - 15? How do we know that it must be y is greater than that then say y must be less than that value?
Director  S
Joined: 12 Nov 2016
Posts: 701
Location: United States
Schools: Yale '18
GMAT 1: 650 Q43 V37 GRE 1: Q157 V158 GPA: 2.66
Re: In the xy-plane, triangular region S is bounded by the lines x=0,y=0,  [#permalink]

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Bunuel wrote:
In the xy-plane, triangular region S is bounded by the lines x=0,y=0, and 3x−2y=30. Which of the following points does NOT lie inside region S?

A. (1,−1)
B. (2,−10)
C. (3,−13)
D. (5,−5)
E. (8,−2)

@Bunuel- although we can diagram this problem, calculate the x and y intercepts which would inevitably allow us to find the answer- can we not just reason that because we are given the hypotenuse of the triangle in this particular form 3x-2y=30 that no set of coordinates within the triangle will result in a number greater than 30? And when I say hypotenuse of the triangle I want to note I am not trying to make a generalization- there was another similar problem in ps made by manhattan in which the triangle in the plane had a x line and y line and an equation by the side it formed by connecting to the x and y lines turned out not to be the hypotenuse.
Intern  B
Joined: 20 Jun 2017
Posts: 7
Re: In the xy-plane, triangular region S is bounded by the lines x=0,y=0,  [#permalink]

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Hi,
I do'nt understand why the point should satisfy the line eqn 3x-2y < = 30 and not required to satify x <= 0 and y <= 0 as both are the eqn of lines that make the triangle. I don'nt know the concept or logic applied . kindly help me with the basic concepts here Bunuel
TIA
Manager  B
Joined: 31 Dec 2018
Posts: 114
Location: India
Re: In the xy-plane, triangular region S is bounded by the lines x=0,y=0,  [#permalink]

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I have a different approach !

As per the graph mentioned by [#permalink] we are clear with x and y intercepts.
the equation of the line becomes:
y=(3/2)x -15

Slope of the line becomes 3/2 or 1.5
this means that when we move 1 unit in x we lose 1.5 units of y

Coming back to the question,
All options have valid x-axis values.
So, if we move 3 units in 3 we will lose 4.5 units of y intercept ie. y will be 15-4.5 = 10.5
But y-axis value is mentioned as -13, this shows that the point lies outside the region S.

Hope this helps ! Re: In the xy-plane, triangular region S is bounded by the lines x=0,y=0,   [#permalink] 23 Apr 2019, 04:32
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